Lerch's $\Phi$ and the Polylogarithm at the Negative Integers

TL;DR

This paper derives simple closed-form expressions for the Lerch $\

Contribution

It introduces new closed-form formulas for the Lerch $\

Findings

Closed-form formula for Lerch $\

Alternative formulas for derivatives of trigonometric functions

Extension of Hurwitz zeta function formulas to the complex half-plane

Abstract

At the negative integers, there is a simple relation between the Lerch $\Phi$ function and the polylogarithm. Starting from that relation and a formula for the polylogarithm at the negative integers known from the literature, we can deduce a simple closed formula for the Lerch $\Phi$ function at the negative integers, where the Stirling numbers of the second kind are not needed. Leveraging that finding, we also produce alternative formulae for the $k$-th derivatives of the cotangent and cosecant (ditto, tangent and secant), as simple functions of the negative polylogarithm and Lerch $\Phi$, respectively, which is evidence of the importance of these functions (they are less exotic than they seem). Lastly, we extend formulae for the Hurwitz zeta function only valid at the positive integers to the complex half-plane using this novelty.